Prey Model of Security Forces Versus Criminals in a Contemporary Ghanaian Community
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KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY, KUMASI PREDATOR - PREY MODEL OF SECURITY FORCES VERSUS CRIMINALS IN A CONTEMPORARY GHANAIAN COMMUNITY By AKUAMOAH WORLANYO SAVIOUR (B.ED. Mathematics) A THESIS SUBMITTED TO THE DEPARTMENT OF MATHEMATICS, KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF THE DEGREE OF M.PHIL APPLIED MATHEMATICS October 9, 2014 Declaration I hereby declare that this submission is my own work towards the award of the M. Phil degree and that, to the best of my knowledge, it contains no material previously published by another person nor material which had been accepted for the award of any other degree of the university, except where due acknowledge- ment had been made in the text. Akuamoah Worlanyo Saviour ....................................... Student Signature Date Certified by: Dr. F.T Oduro ....................................... Supervisor Signature Date Certified by: Prof. S.K. Amponsah ....................................... Head of Department Signature Date i Dedication This work is dedicated to my sweet mother Mabel Awalime, who worked assidu- ously to make me what I am now. ii Abstract A Predator prey model is developed from the population dynamics of security forces versus criminals in a contemporary community. Using five major crime data covering the period of 2000 to 2010 from Ghana Statistical Service, two models were developed and a local stability analysis of the model after determination of the equilibrium points was investigated. With some assumptions, the parameters of the model are estimated and the simulation of the model for various scenarios using MATLAB is done. These simulations give the typical almost sinusoidal trajectories for both the populations of the security forces and criminals. This appears to confirm that the propagation of security forces and criminals follows the predator - prey model. It also shows that the security forces and criminal activities keeps rising and falling with time. It was observed that, by analysing the same point in sequential phases and finding the time in between them the approximate periods of the phase plane cycle is found to be 12 months. The results obtained suggest that criminal activities persist if not the introduction of voluntary guards. iii Acknowledgements I wish to first and foremost express my profound appreciation and gratitude to the Almighty God for his immeasurable love, inexhaustible grace and divine pro- tection and provision throughout the course of my studies in the University. I also wish to express my deepest appreciation to my supervisor Dr. F.T. Oduro for meticulously reading and supervising and making valuable contributions in the realization of the success of this work. Again, I specially wish to thank Madam Awalime Mabel for her advice, and finan- cial support. In the same vein, I wish to make mention of Prof. S.K. Amposah, Furthermore, I thank all lecturers in the Mathematics Department, KNUST for the immense academic knowledge impacted especially during the course work ses- sions. I also express my deepest gratitude to Mr. Obeng Taiwiah, Prince Harvin, David Yaro, Perpetual Andam, Akuamoah Sena, Obeng Adu Benjamin, Chenyuo Boni- face, Afrifa Twumasi, Samuel Asante Gyamera, Aglo Liticia Edem and others for their support and contributions. Finally, to my wonderful family. iv Contents Declaration .................................. v Dedication ................................... v Acknowledgement .............................. v List of Tables ................................. viii List of Figures ................................ ix 1 Introduction ................................ 1 1.1 Background of the Study . 1 1.2 Statement of the Problem . 4 1.3 Objectives of the Study . 5 1.4 Methodology . 5 1.5 Justification of the Study . 6 1.6 The Study Area . 6 1.7 Organization of the Thesis . 7 2 Literature Review ............................ 8 2.1 Introduction . 8 2.2 Crime Overview . 8 2.3 Predator - Prey Model . 14 2.3.1 Predator - Prey Model: Lotka - Volterra Systems . 20 2.4 Concluding Remarks . 21 3 MODEL FORMULATION ...................... 22 v 3.1 Introduction . 22 3.2 Definitions . 22 3.2.1 Differential Equation . 22 3.2.2 Ordinary and Partial Differential Equations . 23 3.2.3 Steady state . 23 3.2.4 Stability of equilibrium point . 23 3.2.5 Unstability of equilibrium point . 23 3.2.6 Asymptotically stable . 24 3.2.7 Linearisation and stability . 24 3.2.8 The Phase Plane . 25 3.2.9 Example 1(Matrices with real distinct eigenvalues of the same sign) . 28 3.2.10 Example II (Matrices with Real eigenvalues of opposite signs) 29 3.2.11 Example III (Matrices with complex conjugate eigenvalues of non zero real parts) . 29 3.2.12 Example IV ( Purely imaginary eigenvalues) . 30 3.2.13 Example V (matrices with repeated eigenvalues) . 31 3.2.14 Liapunov function . 32 3.2.15 Malthusian Model . 34 3.3 Predator - Prey model of Security forces versus Criminals . 34 3.3.1 Model Assumptions . 34 3.4 Model A . 35 3.4.1 Model formulation . 35 3.4.2 Analysis of the model . 37 3.4.3 The equilibrium points . 40 3.4.4 Stability of equilibrium point . 41 3.4.5 Phase path to the system . 42 3.5 Model B . 47 3.5.1 Introduction of Volunteer Guards . 47 vi 3.6 Analysis of Model B . 47 3.6.1 The equilibrium points . 47 3.6.2 Stability of equilibrium points . 48 4 Analysis .................................. 51 4.1 Introduction . 51 4.2 Display of Data . 51 4.3 Parameter Determination . 51 4.4 Numerical Results . 52 4.4.1 Model A . 52 4.4.2 Model B . 52 4.5 Simulation . 53 5 Conclusion ................................. 58 5.1 Recommendation . 59 References ................................... 62 Appendix A .................................. 63 Appendix B .................................. 64 vii List of Tables 3.1 Parameters and their definitions . 35 3.2 Parameters and their definitions . 47 4.1 Parameters and Their Values . 52 viii List of Figures 3.1 Stable centre . 23 3.2 Unstable . 24 3.3 Saddle . 29 3.4 Stable spiral sink . 30 3.5 Stable centre . 31 3.6 Unstable star source . 32 3.7 ..................................... 33 3.8 Proportionate growth rate of criminals as a function of the popu- lation of security forces . 39 3.9 Proportionate growth rate of the security forces as a function of the population of criminals . 40 3.10 . 43 3.11 . 44 3.12 . 44 3.13 A periodic graph of the security forces and the criminals . 44 3.14 Sample Lotka-Volterra Phase Portrait . 45 3.15 . 46 4.1 TRAJECTORIES 0F MODEL A . 56 4.2 PHASE PORTRAIT OF MODEL A . 56 4.3 TRAJECTORIES 0F MODEL B . 57 4.4 PHASE PORTRAIT OF MODEL B . 57 5.1 Criminal Data From 2000 to 2010 . 63 ix Chapter 1 Introduction 1.1 Background of the Study The word crime, from the root of latin cerno¯ means "I decide, I give judgement". According to Fafa (2010), crime is the act of breaking rule(s) or regulation(s) for which a governing authority (via mechanisms such as legal system) can ultimately prescribe a conviction. Crime has also been defined in social or non-legal terms. The social definition of crime is that it is behaviour or an activity that offends the social code of a particular community. Mower (1959) has defined it as "an anti-social act". Karen (1956), has explained it as "an act or a failure to act that is considered to be so detrimental to the well-being of a society, as judged by its prevailing standards, that action against it cannot be entrusted to private initiative or to haphazard methods but must be taken by an organised society in accordance with tested procedures." Sellin (1970) has described crime as "violation of con- duct norms of the normative groups". Clinard (1957) has, however, maintained that all deviations from social norms are not crimes. He talks of three types of deviation: (i) tolerated deviation (ii) deviation which is mildly disapproved and (iii) deviation which is strongly disapproved. He perceives the third type of de- viation as crime. Based on the above definitions, crime can simply be defined as a deviant behaviour that violates prescribed norms or values which is frowned upon by society. A criminal is someone who has committed (or been legally convicted of) a crime. 1 Criminals are common to all societies. To fight against them, contemporary com- munities takes different security measures such as, bringing security forces into the community. Several forms of criminal activities occur in societies which may be described as major or minor crime by virtue of their nature and impact. These are cate- gorised and defined by Levitt (1996) as follows: Robbery- The taking or attempting to take anything of value from the care, cus- tody or control of a person or persons by force, or threat of force, or violence and or by putting the victim in fear Burglary- Entering a building unlawfully with intent to commit a felony or to steal valuable property. Motor Vehicle Theft- The theft or attempted theft of a motor vehicle Larceny - The unlawful taking of property from possession of another. Examples are thefts of bicycles or auto mobile accessories, shoplifting, pocket - picking, or the stealing of any property or article which is not taken by force and violence or by fraud, attempted larcenies are included. Embezzlement, "con" games, forgery and worthless checks are excluded. Aggravated Assault- An unlawful attack by one person upon another for the pur- pose of inflicting severe or aggravated bodily injury. This type of assault usually is accompanied by the use of a weapon or by means likely to produce death or great bodily harm. Simple assaults are excluded Levitt (1996). Murder- The unlawful killing of one human by another, especially with premedi- tated malice source. Forcible Rape- The carnal knowledge of an individual against his or her will.